Question

assume that 0<x<pi/2 and 0<y<pi/2. find the exact value of sin(x+y) if cosx=8/17 and siny= 12/37​

Answer 1

Answer:

621 / 629

Step-by-step explanation:

We are given the following information -

cos x = $\frac{8}{17}$, and sin y = $\frac{12}{37}$

Respectively we can use the following information -

sin( x + y ) = sin x ( cos y ) + sin y ( cos x ),

$cos^2x + sin^2x = 1,\\cos^2y + sin^2y = 1$

Knowing that cos^2x  + sin^2x = 1, cos^2y  + sin^2y = 1, we can calculate the value of sin x and cos y, plugging it into the first bit " sin( x + y ) = sin x ( cos y ) + sin y ( cos x ) "

$sin^2x = 1 - cos^2x,\\sin^2x = 1 - ( 8 / 17 )^2,\\sin^2x = 15^2 / 17^2\\----------------\\sin ( x ) = 15 / 17$

Respectively cos y should be 35 / 37 -

$cos^2y = 1 - sin^2y,\\cos^2y = 1 - ( 12 / 37 )^2,\\cos^2y = 35^2 / 37^2\\----------------\\cos ( y ) = 35 / 37$

Thus,

sin( x + y ) = ( 15 / 17 ) * ( 35 / 37 ) + ( 12 / 37 ) * ( 8 / 17 ),

sin( x + y ) = 621 / 629

Hope that helps!